What is Miller's Magical Number Seven? How has it influenced the way we teach languages?

The psychologist George A Miller had an incalculable influence on the new academic discipline of cognitive science and the still disputed notion of psycho-linguistics. His research focussed on working memory - the way the mind processes information.  He observed that the brain
encountering the unfamiliar, could absorb roughly seven new things at a time.When asked to repeat a random list of letters, words or numbers, he wrote, people got stuck “somewhere in the neighborhood of seven.”
Some people could recall nine items on the list, some fewer than seven. But regardless of the things being recalled — colors and tastes, numbers with decimals, numbers without decimals, consonants, vowels — seven was the statistical average for short-term storage. (Long-term memory, which followed another cognitive formula, was virtually unlimited.) From NYT obituary 02/08/2012

Why is Miller's work important for language teachers?
In practical terms this means that the mind best processes new data by what Miller termed  'chunking' or breaking down into smaller units
Telephone numbers and Social Security numbers are good examples. The number (212) 374-5278, written in three chunks, is a lot easier to cope with than 2123745278.
This has profound implications for the teaching of languages. It demonstrates that trying to introduce too many new words or concepts at once is self-defeating. And it suggests the importance of context - new word are learned more easily when they build on pre-existing vocabulary knowledge
Words are fantastically effective chunking devices. Suppose you put a single item into your working memory—say, “Pasteur.” So long as you hold in your long-term memory a lot of associations with that name, you don’t need to dredge them up and try to cram them into your working memory.  From A Wealth of Words 
Miller's research helped provide a platform for new approaches to language learning - Krashen's 'affective learning' and content-based learning for example.